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The geometric mean of two positive numbers is never greater than the arithmetic mean. [3] So the geometric means are an increasing sequence g 0 ≤ g 1 ≤ g 2 ≤ ...; the arithmetic means are a decreasing sequence a 0 ≥ a 1 ≥ a 2 ≥ ...; and g n ≤ M(x, y) ≤ a n for any n. These are strict inequalities if x ≠ y. M(x, y) is thus a ...
The geometric mean is more appropriate than the arithmetic mean for describing proportional growth, both exponential growth (constant proportional growth) and varying growth; in business the geometric mean of growth rates is known as the compound annual growth rate (CAGR). The geometric mean of growth over periods yields the equivalent constant ...
Early studies showed that individuals who swam in waters with geometric mean coliform densities above 2300/100 mL for three days had higher illness rates. [6] In the 1960s, these numbers were converted to fecal coliform concentrations assuming 18 percent of total coliforms were fecal.
The arithmetic mean, or less precisely the average, of a list of n numbers x 1, x 2, . . . , x n is the sum of the numbers divided by n: + + +. The geometric mean is similar, except that it is only defined for a list of nonnegative real numbers, and uses multiplication and a root in place of addition and division:
The fecal coliform assay is intended to be an indicator of fecal contamination; more specifically of E. coli which is an indicator microorganism for other pathogens that may be present in feces. Presence of fecal coliforms in water may not be directly harmful, and do not necessarily indicate the presence of feces. [1]
In mathematics, the QM-AM-GM-HM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean, geometric mean, arithmetic mean, and quadratic mean (also known as root mean square). Suppose that ,, …, are positive real numbers. Then
The presence of indicator bacteria is measured in a variety of ecosystems and sometimes alongside other measurements. In the Great Lakes, a study was conducted testing for both fecal indicator bacteria (FIB) concentrations and pathogen gene markers. [7] The FIB measured in this study included fecal coliform bacteria, E. coli, and enterococci. [7]
Nomograms to graphically calculate arithmetic (1), geometric (2) and harmonic (3) means, z of x=40 and y=10 (red), and x=45 and y=5 (blue) Of all pairs of different natural numbers of the form ( a , b ) such that a < b , the smallest (as defined by least value of a + b ) for which the arithmetic, geometric and harmonic means are all also ...